x - y + 2z + w = +4 -2x + 3y = -4 x + y + z - 4w = +3 Rewrite linear system as matrix
p x + y+ 2z Subject to x+ 2y + 2z 60 2x +y + 3z 60 +3y+ 6z s 60 Maximize x, y,z 2 0 p x + y+ 2z Subject to x+ 2y + 2z 60 2x +y + 3z 60 +3y+ 6z s 60 Maximize x, y,z 2 0
5. For the system, 4x + y + 2z = 1 2x + 3y + 4z = -5 x – y +3z = 3 Find the rank of the coefficient matrix by calculating the determinant. Use Cramer's theorem to find the solution of this system. (10 points) 6. Find the inverse of the following matrix using Gauss-Jordan method. Verify your result by computing the inverse using the method of determinants. (10 points) 1 2 4 2 4 2 1] 1...
3. Suppose x,y,z satisfy the competing species equations <(6 - 2x – 3y - 2) y(7 - 2x - 3y - 22) z(5 - 2x - y -22) (a) (6 points) Find the critical point (0,Ye, ze) where ye, we >0, and sketch the nullclines and direction arrows in the yz-plane. (b) (6 points) Determine if (0, yc, ze) is stable. (c) (8 points) Determine if the critical point (2,0,0) is stable, where I > 0.
3- a) Assume S = {(x,y,z, 1) : 2x - 3y + z-t=0). Show why S is a subspace of R. b) Given a matrix A- 1 -1 2 1 show a basis of the null space: N(A). 01-11
3. Suppose x, y, z satisfy the competing species equations 2(6 - 2x - 3y - 2) y(7 - 2.0 - 3y - 22) z(5 – 2x - y -22) (a) (6 points) Find the critical point (0, yc, ze) where yc, ze >0, and sketch the nullclines and direction arrows in the yz-plane. (b) (6 points) Determine if (0, yc, ze) is stable. (c) (8 points) Determine if the critical point (1,0,0) is stable, where 8c > 0.
3. Suppose x, y, z satisfy the competing species equations 2(6 - 2x - 3y - 2) y(7 - 2.0 - 3y - 22) z(5 – 2x - y -22) (a) (6 points) Find the critical point (0, yc, ze) where yc, ze >0, and sketch the nullclines and direction arrows in the yz-plane. (b) (6 points) Determine if (0, yc, ze) is stable. (c) (8 points) Determine if the critical point (1,0,0) is stable, where 8c > 0.
(1 point) Consider the vector field F(x, y, z) = (2z + 3y)i + (2z + 3x)j + (2y + 2x)k. a) Find a function f such that F = Vf and f(0,0,0) = 0. f(x, y, z) = b) Suppose C is any curve from (0,0,0) to (1,1,1). Use part a) to compute the line integral / F. dr. (1 point) Verify that F = V and evaluate the line integral of F over the given path: F =...
QUESTION 3 Given the following equations: 2x + 3y + z = 13 3x – 4y + 2z=3 x + y + z = 6 Determine x, y, and z. X = , y = and z =
Construct and evaluate a surface integral that represents the work done by the vector field F(x, y, z)-(x, 2z, 3y), around the triangular section of the plane 2x + y + z traversed counterclockwise. 8 in the first octant Construct and evaluate a surface integral that represents the work done by the vector field F(x, y, z)-(x, 2z, 3y), around the triangular section of the plane 2x + y + z traversed counterclockwise. 8 in the first octant